Software Verification and System Assurance

Transcription

1 Software Verification and System Assurance John Rushby Based on joint work with Bev Littlewood (City University UK) Computer Science Laboratory SRI International Menlo Park CA USA John Rushby, SR I Verification & Assurance 1

2 Introduction Verification was the original goal of formal methods Proving correctness of programs Nowadays, overtaken by more pragmatic goals Generating test cases, synthesizing monitors, bug finding Verifying limited properties: static analysis, extended type checking Also, the whole idea of verification has come into question Proving the absence of errors is an overly strong claim Lots of caveats about assumptions, fidelity of models etc. Plus, software is usually only part of a larger system And it s the system we care about John Rushby, SR I Verification & Assurance 2

3 Software Correctness vs. System Claims The top-level requirements for most complex systems are stated quantitatively E.g., no catastrophic failure in the lifetime of all airplanes of one type And these lead to probabilistic requirements for software-intensive subsystems E.g., probability of failure in flight control less than 10 9 per hour But formal methods in general And formal verification in particular Are about correctness... an absolute notion How do we connect the absolute claims of formal verification for software with probabilistic requirements at the system level? John Rushby, SR I Verification & Assurance 3

4 Software Reliability Software contributes to system failures through faults in its requirements, design, implementation bugs A bug that leads to failure is certain to do so whenever it is encountered in similar circumstances There s nothing probabilistic about it Aaah, but the circumstances of the system are a stochastic process So there is a probability of encountering the circumstances that activate the bug Hence, probabilistic statements about software reliability or failure are perfectly reasonable Typically speak of probability of failure on demand (pfd), or failure rate (per hour, say) John Rushby, SR I Verification & Assurance 4

5 Measuring/Predicting Software Reliability For pfds down to about 10 4, it is feasible to measure software reliability by statistically valid random testing But 10 9 would need 114,000 years on test So how do we establish that a piece of software is adequately reliable for a system that requires, say, 10 6? Most standards for system safety (e.g., IEC 61508, DO178B) require you to show that you did a lot of V&V e.g., 57 V&V objectives at DO178B Level C (10 5 ) And you have to do more for higher levels 65 objectives at DO178B Level B (10 7 ) 66 objectives at DO178B Level A (10 9 ) John Rushby, SR I Verification & Assurance 5

6 Does More Correct Mean More Reliable? These V&V objectives are all about correctness Requirements tracing, testing etc. More V&V objectives might make the software more correct but what does that have to do with reliability? And what does more correct mean anyway? John Rushby, SR I Verification & Assurance 6

7 Possibly Perfect Software Instead of correct software Which is about conformance with specification We ll speak of perfect software Software that will never experience a failure in operation, no matter how much operational exposure it has You might not believe a given piece of software is perfect But you might concede it has a possibility of being perfect And the more V&V it has had, the greater that possibility So let s speak of a probability of perfection Think of all the software that might have been developed by comparable engineering processes to solve the same design problem as the software at hand The probability of perfection is then the probability that any software randomly selected from this class is perfect John Rushby, SR I Verification & Assurance 7

8 Probabilities of Perfection and Failure Probability of perfection relates to correctness-based V&V And it also relates to reliability: By the formula for total probability P(s/w fails [on a randomly selected demand]) (1) = P(s/w fails s/w perfect) P(s/w perfect) + P(s/w fails s/w imperfect) P(s/w imperfect). The first term in this sum is zero, because the software does not fail if it is perfect Can then, very conservatively, assume that the software always fails if it imperfect, so that the first factor in the second term becomes 1 Hence, very crudely, and very conservatively, P(software fails) < P(software imperfect) (2) John Rushby, SR I Verification & Assurance 8

9 Two Channel Systems Many safety-critical systems have two (or more) diverse channels E.g., nuclear shutdown, flight control One operational channel does the business A simpler channel provides a backup or monitor Cannot simply multiply the pfds of the two channels to get pfd for the system Failures are unlikely to be independent Failure of one channel suggests this is a difficult case, so failure of the other is more likely Infeasible to measure amount of dependence John Rushby, SR I Verification & Assurance 9

10 Two Channel Systems and Possible Perfection But if the second channel is possibly perfect Its imperfection is conditionally independent of failures in the first channel Hence, system pfd is conservatively bounded by product of pfd of first channel and probability of imperfection of the second Joint work with Bev Littlewood (reported elsewhere) Who originated the idea of possible perfection May provide justification for some of the architectures suggested in ARP 4754 e.g., 10 9 system made of Level C operational channel and Level A monitor John Rushby, SR I Verification & Assurance 10

11 Aside: Monitors Do Fail Fuel emergency on Airbus A , G-VATL, 8 February 2005 Type 1 failure EFIS Reboot during spin recovery on Airbus A300 (American Airlines Flight 903), 12 May 1997 Type 2 failure Current proposals are for formally synthesized/verified monitors for properties in the safety case John Rushby, SR I Verification & Assurance 11

12 Aleatory and Epistemic Uncertainty Aleatory or irreducible uncertainty is uncertainty in the world e.g., if I have a biased coin with P(heads) = p h, I cannot predict exactly how many heads will occur in 100 trials because of randomness in the world Frequentist interpretation of probability needed here Epistemic or reducible uncertainty is uncertainty about the world e.g., if I give you the biased coin, you will not know p h ; you can estimate it, and can try to improve your estimate by doing experiments, learning something about its manufacture, the historical record of similar coins etc. Frequentist and subjective interpretations OK here John Rushby, SR I Verification & Assurance 12

13 Aleatory and Epistemic Uncertainty in Models In much scientific modeling, the aleatory uncertainty is captured conditionally in a model with parameters And the epistemic uncertainty centers upon the values of these parameters As in the coin tossing example Analysis in (1) was aleatory, with parameters p np probability the software is imperfect p fnp probability that it fails, if it is imperfect P(software fails) < p fnp p np John Rushby, SR I Verification & Assurance 13

14 Epistemic Estimation To apply this result, we need to assess values for p fnp and p np These are most likely subjective probabilities i.e., degrees of belief Beliefs may not be independent So will be represented by some joint distribution F(p fnp, p np ) Probability of system failure will be given by the Riemann-Stieltjes integral p fnp p np df(p fnp, p np ). (3) 0 p fnp 1 0 pnp 1 If beliefs can be separated F factorizes as F(p fnp ) F(p np ) And (3) becomes P fnp P np Where these are the means of the posterior distributions representing the assessor s beliefs about the two parameters John Rushby, SR I Verification & Assurance 14

15 Formal Verification and the Probability of Perfection We want to assess P np Context is likely a safety case in which claims about a system are justified by an argument based on evidence about the system and its development Suppose part of the evidence is formal verification i.e., what is the probability of perfection of formally verified software? Let s consider where formal verification can go wrong John Rushby, SR I Verification & Assurance 15

16 The Basic Requirements For The Software Are Wrong This error is made before any formalization It seems to be the dominant source of errors in flight software But monitoring and backup software is built to requirements taken directly from the safety case If these are wrong, we have big problems In any case, it s not specific to formal verification So we ll discount this concern John Rushby, SR I Verification & Assurance 16

17 The Requirements etc. are Formalized Incorrectly Could also be the assumptions, or the design Formalization may be inconsistent i.e., meaningless Many Z specs are like this Can be eliminated using constructive specifications In a tool-supported framework That guarantees conservative extension But that s not always appropriate Prefer to state assumptions as axioms Consistency can then be guaranteed by exhibiting a constructive model (interpretation) PVS can do this So we can eliminate this concern John Rushby, SR I Verification & Assurance 17

18 The Requirements etc. are Formalized Incorrectly (ctd.) Formalization may be consistent, but wrong In my experience, this is the dominant source of errors in formal verification There are papers on errors in my specifications Formal specifications that have not been subjected to analysis are no more likely to be correct than programs that have never been run In fact, less so: engineers have better intuitions about programs than specifications Should challenge formal specifications Prove putative theorems Get counterexamples for deliberately false conjectures Directly execute them on test cases Social process operates on widely used theories John Rushby, SR I Verification & Assurance 18

19 The Requirements etc. are Formalized Incorrectly (ctd. 2) Even if a theory or specification is formalized incorrectly, it does not necessarily invalidate all theorems that use it Only if the verification actually exploits the incorrectness will the validity of the theorem be in doubt Even then, it could still be true, but unproven Some verification systems identify all the axioms and definitions on which a formally verified conclusion depends PVS does this If these are correct, then logical validity of the verified conclusion follows by soundness of the verification system So can apply special scrutiny to them John Rushby, SR I Verification & Assurance 19

20 The Formal Specification and Verification is Discontinuous or Incomplete Discontinuities arise when several analysis tools are applied in the same specification e.g., static analyzer, model checker, timing analyzer Concern is that different tools ascribe different semantics Increasing issue as specialized tools outstrip monolithic ones Need integrating frameworks such as a tool bus Most significant incompleteness is generally the gap between the most detailed model and the real thing Algorithms vs. code, libraries, OS calls That s one reason why we still need testing Driven from the formal specification Cf. penetration tests for secure systems: probe the assumptions John Rushby, SR I Verification & Assurance 20

21 Unsoundness In the Verification System All verification systems have had soundness bugs But none have been exploited to prove a false theorem Many efforts to guarantee soundness are costly e.g., reduction to elementary steps, proof objects What does soundness matter if you cannot do the proof? A better approach is KOT: the Kernel Of Truth (Shankar) A ladder of increasingly powerful verified checkers Untrusted prover leaves a trail, blessed by verified checker More powerful checkers guaranteed by one-time check of its verification by the one above The more powerful the verified checker, the more economical the trail can be (little more than hints) John Rushby, SR I Verification & Assurance 21

22 KOT: A Ladder of Verified Checkers Untrusted Frontline Verifier Hints Verified Offline Verifier Certificates Verified Checker Proofs Trusted Proof Kernel Shankar and Marc Vaucher have verified a modern SAT solver that is executable (modulo lacunae in the PVS evaluator) John Rushby, SR I Verification & Assurance 22

23 Application Suppose we need P np of 10 4 Bulk of this budget should be divided between incorrect formalization and incompleteness of the formal analysis, with small fraction allocated to unsoundness of verification system Through sufficiently careful and comprehensive formal challenges, it is plausible an assessor can assign a subjective posterior probability of imperfection on the order of 10 4 to the formal statements on which a formal verification depends Through testing and other scrutiny, a similar figure can be assigned to the probability of imperfection due to discontinuities and incompleteness in the formal analysis By use of a verification system with a trusted or verified kernel, or trusted, verified, or diverse checkers, assessor can assign probability of 10 5 or smaller that the theorem prover incorrectly verified the theorems that attest to perfection John Rushby, SR I Verification & Assurance 23

24 Discussion These numbers are feasible and plausible Formal methods and their tools do not need to be held to (much) higher standards than the systems they assure But what are we to do about single channel systems that require 10 9? Type 2 failures of monitors (incorrect activation) may be in this class Topic for investigation and discussion whether such assessments could be considered feasible and credible John Rushby, SR I Verification & Assurance 24

25 Conclusion Probability of perfection is a radical and valuable idea Provides the bridge between correctness-based verification activities and probabilistic claims needed at the system level Relieves formal verification, and its tools, of the burden of absolute perfection John Rushby, SR I Verification & Assurance 25